×
Przytycki, Feliks

Ergodicity of toral linked twist mappings. (English) Zbl 0531.58031

Ann. Sci. Éc. Norm. Supér. (4) 16, 345-354 (1983).
Linked twist mappings occur in a variety of settings in dynamical systems. For example, they arise naturally in the Thurston theory of pseudo-Anosov maps as well as in mechanics in the Stormer problem, the problem of the motion of a charged particle in the field of a magnetic dipole. Linked twist maps on tori are described by composing two Dehn twists in complementary annuli in the torus. These maps clearly preserve Lebesgue measure. If the twists are all in the ”right” direction, then R. Burton and R. W. Easton have proved that the map is ergodic [Lect. Notes Math. 819, 35-49 (1980; Zbl 0451.58023)]. The purpose of this paper is to extend these results to the case of ”twists” or ”shears” in different directions. The author shows that as long as these shears are not too strong, the map is still ergodic and in fact satisfies the Bernoulli property.
Reviewer: R.Devaney

Cited in 17 Documents

MSC:

37A99 Ergodic theory

Keywords:

pseudo-Anosov maps; Stormer problem; Bernoulli property

Citations:

Zbl 0451.58023
Cite Review PDF
Full Text: DOI Numdam EuDML

References:

[1] R. BOWEN , On Axiom A Diffeomorphisms (Proc. C.B.M.S. Regional Conf. Ser. Math., N^\circ 35, Amer. Math. Soc., Providence R. I.). MR 58 #2888 | Zbl 0383.58010 · Zbl 0383.58010
[2] R. BURTON and R. EASTON , Ergodicity of Linked Twist Mappings (Global Theory of Dynamical Systems, Proc., Northwestern 1979 , Lecture Notes in Math., n^\circ 819, pp. 35-49). MR 82b:58046 | Zbl 0451.58023 · Zbl 0451.58023
[3] R. DEVANEY , Linked Twist Mappings are Almost Anosov (Global theory of Dynamical Systems, Proc. Northwestern 1979 , Lecture Notes in Math., n^\circ 819, pp. 121-145). MR 82f:58070 | Zbl 0448.58018 · Zbl 0448.58018
[4] R. EASTON , Chain Transitivity and the Domain of Influence of an Invariant Set (Lecture Notes in Math., n^\circ 668, pp. 95-102). MR 80j:58051 | Zbl 0393.54027 · Zbl 0393.54027
[5] A. KATOK , Ya. G. SINAI and A. M. STEPIN , Theory of Dynamical Systems and General Transformation Groups with Invariant Measure (I togi Nauki i Tekhniki, Matematicheskii Analiz, Vol. 13, 1975 , pp. 129-262 (In Russian). English translation : J. of Soviet Math., Vol. 7, N^\circ 6, 1977 , pp. 974-1065). Zbl 0399.28011 · Zbl 0399.28011 · doi:10.1007/BF01223133
[6] A. KATOK and J.-M. STRELCYN , Invariant Manifolds for Smooth Maps with Singularities I . Existence, II. Absolute Continuity, preprint, The Pesin Entropy Formula for Smoth Maps with Singularities, preprint.
[7] M. WOJTKOWSKI , Linked Twist Mappings Have the K-Property (Nonlinear Dynamics, International Conference, New York 1979 , pp. 66-76). Zbl 0475.58008 · Zbl 0475.58008
[8] M. WOJTKOWSKI , A Model Problem with the Coexistence of Stochastic and Integrable Behaviour (Comm. Math. Phys., Vol. 80, N^\circ 4, 1981 , pp. 453-464). Article | MR 83a:28023 | Zbl 0473.28006 · Zbl 0473.28006 · doi:10.1007/BF01941656
[9] YA. B. PESIN , Lyapunov Characteristic Exponents and Smooth Ergodic Theory (Uspehi Mat. Nauk., Vol. 32, n^\circ 4 (196), 1977 , pp. 55-112. English translation : Russian Math. Surveys, Vol. 32, No. 4, 1977 , pp. 55-114). Zbl 0383.58011 · Zbl 0383.58011 · doi:10.1070/RM1977v032n04ABEH001639
[10] W. THURSTON , On the Geometry and Dynamics of Diffeomorphisms of Surfaces , I, preprint. · Zbl 0674.57008
[11] F. PRZYTYCKI , Linked Twist Mappings : Ergodicity , preprint I.H.E.S., February 1981 .
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.